On Maximality of Quasimonotone Operators

We introduce the notion of quasimonotone polar of a multivalued operator, in a similar way as the well-known monotone polar due to Martínez-Legaz and Svaiter. We first recover several properties similar to the monotone polar, including a characterization in terms of normal cones. Next, we use it to analyze certain aspects of maximal (in the sense of graph inclusion) quasimonotonicity, and its relation to the notion of maximal quasimonotonicity introduced by Aussel and Eberhard. Furthermore, we study the connections between quasimonotonicity and Minty Variational Inequality Problems and, in particular, we consider the general minimization problem. We conclude by characterizing the maximal quasimonotonicity of operators defined in the real line.